Controllability in the max-algebra

summary:We are interested here in the reachability and controllability problems for DEDS in the max-algebra. Contrary to the situation in linear systems theory, where controllability (resp observability) refers to a (linear) subspace, these properties are essentially discrete in the $\max $-linear dynamic system. We show that these problems, which consist in solving a $\max $-linear equation lead to an eigenvector problem in the $\min $-algebra. More precisely, we show that, given a $\max $-linear system, then, for every natural number $k\ge 1\,$, there is a matrix $\Gamma _k$ whose $\min $-eigenspace associated with the eigenvalue $1$ (or $\min $-fixed points set) contains all the states which are reachable in $k$ steps. This means in particular that if a state is not in this eigenspace, then it is not controllable. Also, we give an indirect characterization of $\Gamma _k$ for the condition to be sufficient. A similar result also holds by duality on the observability side